The middle hedgehog of a planar convex body

TL;DR

This paper proves that most planar convex bodies have infinitely many convexity points, using the concept of the middle hedgehog and properties of affine diameters.

Contribution

It introduces the middle hedgehog of a convex body and demonstrates that typical planar convex bodies have infinitely many convexity points.

Findings

Typical convex bodies have infinitely many convexity points.

The convex hull of the middle hedgehog has infinitely many exposed points.

The proof uses Baire category arguments.

Abstract

A convexity point of a convex body is a point with the property that the union of the body and its reflection in the point is convex. It is proved that in the plane a typical convex body (in the sense of Baire category) has infinitely many convexity points. The proof makes use of the `middle hedgehog' of a planar convex body $K$, which is the curve formed by the midpoints of all affine diameters of $K$. The stated result follows from the fact that for a typical planar convex body the convex hull of the middle hedgehog has infinitely many exposed points.